Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,21,24,37)(12,17)(13,27)(14,19)(15,20)(16,23,32,40)(18,26)(22,35,25,38)(28,31,39,33)(29,36)(30,34);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,16,20,32)(12,25,14,22)(13,28,30,39)(17,33,19,31)(18,35,36,38)(21,27,37,34)(23,26,40,29);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,30,29)(16,31,32,33)(18,34,36,27)(21,35,37,38)(23,39,40,28)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128809, #128837, #129770, … |
| maximal normal subgroup of | #128809, #128837, #129770, … |
| maximal quotient of | #122546, #122548, #122551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4694);